This book treats the very special and fundamental mathematical properties of a family of Gaussian (or normal) random variables. Such random variables have many applications in probability theory, statistics and theoretical physics. The book concentrates on the mathematical structures common to all these applications. This will be an excellent resource for all researchers whose work involves random variables.
Table of Contents1. Gaussian Hilbert spaces; 2. Wiener chaos; 3. Wick products; 4. Tensor products and Fock spaces; 5. Hypercontractivity; 6. Distributions of variables with finite chaos expansions; 7. Stochastic integration; 8. Gaussian stochastic processes; 9. Conditioning; 10. Limit theorems for generalized U-statistics; 11. Applications to operator theory; 12. Some operators from quantum physics; 13. The Cameron-Martin shift; 14. Malliavin calculus; 15. Transforms; Appendices.